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Unformatted text preview: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Number Theory 2 The Integers and Division Of course, you already know what the integers are, and what division is However: There are some specific notations, terminology, and theorems associated with these concepts which you may not know. These form the basics of number theory . Vital in many important algorithms today (hash functions, cryptography, digital signatures; in general, online security). 3 The divides operator New notation: 3  12 To specify when an integer evenly divides another integer Read as 3 divides 12 The notdivides operator: 5  12 To specify when an integer does not evenly divide another integer Read as 5 does not divide 12 4 Divides, Factor, Multiple Let a , b Z with a . Def.: a  b a divides b : ( 5 c Z : b=ac ) There is an integer c such that c times a equals b. Example: 3  12 True , but 3  7 False . Iff a divides b , then we say a is a factor or a divisor of b , and b is a multiple of a . Ex.: b is even : 2 b . Is 0 even? Is 4? 5 Results on the divides operator If a  b and a  c, then a  (b+c) Example: if 5  25 and 5  30, then 5  (25+30) If a  b, then a  bc for all integers c Example: if 5  25, then 5  25*c for all ints c If a  b and b  c, then a  c Example: if 5  25 and 25  100, then 5  100 (common facts but good to repeat for background) 6 Divides Relation Theorem: 2200 a , b , c Z : 1. a 0 2. ( a  b a  c ) a  ( b + c ) 3. a  b a  bc 4. ( a  b b  c ) a  c Proof of (2): a  b means there is an s such that b = as , and a  c means that there is a t such that c = at , so b + c = as + at = a ( s + t ), so a ( b + c ) also. Corollary: If a, b, c are integers , such that a  b and a  c , then a  mb + nc whenever m and n are integers . 7 More Detailed Version of Proof Show 2200 a , b , c Z : ( a  b a  c ) a  ( b + c ) . Let a , b , c be any integers such that a  b and a  c , and show that a  ( b + c ) . By defn. of  , we know 5 s : b=as , and 5 t : c=at . Let s , t , be such integers. Then b + c = as + at = a ( s + t ) , so 5 u : b + c = au , namely u = s + t . Thus a ( b + c ) . Divides Relation Corollary: If a, b, c are integers , such that a  b and a  c , then a  mb + nc whenever m and n are integers . Proof: From previous theorem part 3 (i.e., ab abe) it follows that a  mb and a  nc ; again, from previous theorem part 2 (i.e., ( a  b a  c ) a  ( b + c )) it follows that a  mb + nc 9 The Division Algorithm Theorem: Division Algorithm  Let a be an integer and d a positive integer. Then there are unique integers q and r, with 0 r < d, such that a = dq+r....
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